Internal Parametricity for Cubical Type Theory

Cavallo, Evan; Harper, Robert

doi:10.46298/lmcs-17(4:5)2021

Cited by 9 publications

(16 citation statements)

References 21 publications

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“…• We show that this extra structure of parametric right adjoints allows FitchTT to emulate the convenient syntax of handcrafted type theories for internalized parametricity [7,16] and guarded recursion [4,5].…”

Section: Contributionsmentioning

confidence: 99%

“…However, we discover that 'extra-logical' structures found in several prior type theories can be seen as instances of these rules. For example, we show that the extension of a context by an affine dimension variable, as used in internalizing parametricity [7,16], forms a PRA. Furthermore, the 'tick variables' used in clocked type theory [4] can also be seen as arising from a PRA.…”

Section: Substitution and The Fitch Stylementioning

confidence: 99%

“…Some version of Lemma 1 has been shown en passant for all prior Fitch-style calculi in the process of proving the admissibility of substitution [4,9,16,25]. In each case the modal elimination rules can be derived by unfolding the components of the parametric adjunction in the general rule dra/tm/pra-unmod.…”

Section: Parametric Right Adjointsmentioning

confidence: 99%

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Modalities and Parametric Adjoints

Gratzer

Cavallo

Kavvos

et al. 2022

ACM Trans. Comput. Logic

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Birkedal et al. recently introduced dependent right adjoints as an important class of (non-fibered) modalities in type theory. We observe that several aspects of their calculus are left underdeveloped, and that it cannot serve as an internal language. We resolve these problems by assuming that the modal context operator is a parametric right adjoint. We show that this hitherto unrecognized structure is common. Based on these discoveries we present a new well-behaved Fitch-style multimodal type theory, which can be used as an internal language. Finally, we apply this syntax to guarded recursion and parametricity.

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Section: Contributionsmentioning

confidence: 99%

Section: Substitution and The Fitch Stylementioning

confidence: 99%

See 1 more Smart Citation

Modalities and Parametric Adjoints

Gratzer

Cavallo

Kavvos

et al. 2022

ACM Trans. Comput. Logic

Self Cite

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show abstract

“…Abstracting from the semantics of parametricity, it is possible to consider extensions of dependent type theory for internal parametricity that involve connectives for relatedness [18,19,21,31,32,73,98]. Semantic parametricity, especially as embodied in reflexive graphs, can be seen to be a truncation of a much higher-dimensional structure; going one level up, one can consider a reflexive graph enriched in reflexive graphs, but there is no need to stop there.…”

Section: Internal Parametricitymentioning

confidence: 99%

Logical Relations as Types: Proof-Relevant Parametricity for Program Modules

Sterling

Harper

2021

J. ACM

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The theory of program modules is of interest to language designers not only for its practical importance to programming, but also because it lies at the nexus of three fundamental concerns in language design: the phase distinction , computational effects , and type abstraction . We contribute a fresh “synthetic” take on program modules that treats modules as the fundamental constructs, in which the usual suspects of prior module calculi (kinds, constructors, dynamic programs) are rendered as derived notions in terms of a modal type-theoretic account of the phase distinction. We simplify the account of type abstraction (embodied in the generativity of module functors) through a lax modality that encapsulates computational effects, placing projectibility of module expressions on a type-theoretic basis. Our main result is a (significant) proof-relevant and phase-sensitive generalization of the Reynolds abstraction theorem for a calculus of program modules, based on a new kind of logical relation called a parametricity structure . Parametricity structures generalize the proof-irrelevant relations of classical parametricity to proof- relevant families, where there may be non-trivial evidence witnessing the relatedness of two programs—simplifying the metatheory of strong sums over the collection of types, for although there can be no “relation classifying relations,” one easily accommodates a “family classifying small families.” Using the insight that logical relations/parametricity is itself a form of phase distinction between the syntactic and the semantic, we contribute a new synthetic approach to phase separated parametricity based on the slogan logical relations as types , by iterating our modal account of the phase distinction. We axiomatize a dependent type theory of parametricity structures using two pairs of complementary modalities (syntactic, semantic) and (static, dynamic), substantiated using the topos theoretic Artin gluing construction. Then, to construct a simulation between two implementations of an abstract type, one simply programs a third implementation whose type component carries the representation invariant.

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“…Moreover it is known that cubical structures arise when trying to internalize parametricity in type theory. For example [7] gives a model for (the unary variant of) parametricity in (the unary variant of) cubical sets, and [12] shows how parametricity can be internalized orthogonally to univalence, using very similar cubical techniques for both features.…”

Section: Parametricity and Cubical Structurementioning

confidence: 99%

Parametricity and Semi-Cubical Types

Moeneclaey¹

2021

Preprint

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We construct a model of type theory enjoying parametricity from an arbitrary one. A type in the new model is a semi-cubical type in the old one, illustrating the correspondence between parametricity and cubes.Our construction works not only for parametricity, but also for similar interpretations of type theory and in fact similar interpretations of any generalized algebraic theory. To be precise we consider a functor forgetting unary operations and equations defining them recursively in a generalized algebraic theory. We show that it has a right adjoint.We use techniques from locally presentable category theory, as well as from quotient inductive-inductive types.

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Internal Parametricity for Cubical Type Theory

Cited by 9 publications

References 21 publications

Modalities and Parametric Adjoints

Modalities and Parametric Adjoints

Logical Relations as Types: Proof-Relevant Parametricity for Program Modules

Parametricity and Semi-Cubical Types

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